Theorem sstpr 3502

Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
sstpr	⊢ ((((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) ∨ ((A = {𝐷} ∨ A = {B, 𝐷}) ∨ (A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})

Proof of Theorem sstpr

Step	Hyp	Ref	Expression
1		ssprr 3501	. . 3 ⊢ (((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) → A ⊆ {B, 𝐶})
2		prsstp12 3491	. . 3 ⊢ {B, 𝐶} ⊆ {B, 𝐶, 𝐷}
3	1, 2	syl6ss 2934	. 2 ⊢ (((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) → A ⊆ {B, 𝐶, 𝐷})
4		snsstp3 3490	. . . . 5 ⊢ {𝐷} ⊆ {B, 𝐶, 𝐷}
5		sseq1 2943	. . . . 5 ⊢ (A = {𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐷} ⊆ {B, 𝐶, 𝐷}))
6	4, 5	mpbiri 157	. . . 4 ⊢ (A = {𝐷} → A ⊆ {B, 𝐶, 𝐷})
7		prsstp13 3492	. . . . 5 ⊢ {B, 𝐷} ⊆ {B, 𝐶, 𝐷}
8		sseq1 2943	. . . . 5 ⊢ (A = {B, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {B, 𝐷} ⊆ {B, 𝐶, 𝐷}))
9	7, 8	mpbiri 157	. . . 4 ⊢ (A = {B, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
10	6, 9	jaoi 623	. . 3 ⊢ ((A = {𝐷} ∨ A = {B, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
11		prsstp23 3493	. . . . 5 ⊢ {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}
12		sseq1 2943	. . . . 5 ⊢ (A = {𝐶, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}))
13	11, 12	mpbiri 157	. . . 4 ⊢ (A = {𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
14		eqimss 2974	. . . 4 ⊢ (A = {B, 𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
15	13, 14	jaoi 623	. . 3 ⊢ ((A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
16	10, 15	jaoi 623	. 2 ⊢ (((A = {𝐷} ∨ A = {B, 𝐷}) ∨ (A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷})) → A ⊆ {B, 𝐶, 𝐷})
17	3, 16	jaoi 623	1 ⊢ ((((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) ∨ ((A = {𝐷} ∨ A = {B, 𝐷}) ∨ (A = {𝐶, 𝐷} ∨ A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228   ⊆ wss 2894  ∅c0 3201  {csn 3350  {cpr 3351  {ctp 3352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3or 874  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-tp 3358

This theorem is referenced by:  pwtpss  3551